In this paper,we consider the discrete boundary value problem of the type{∆u1=0=∆un-1,∇(t_(k)^(N-1))φ(∆uk))+t_(k)^(N-1)fk(t_(k),u_(k),∆_(uk))=0,2≤k≤n-1,whereφ:(-a,a)→R,0<a<∞,is an increasing homeomorphism ...In this paper,we consider the discrete boundary value problem of the type{∆u1=0=∆un-1,∇(t_(k)^(N-1))φ(∆uk))+t_(k)^(N-1)fk(t_(k),u_(k),∆_(uk))=0,2≤k≤n-1,whereφ:(-a,a)→R,0<a<∞,is an increasing homeomorphism withφ(0)=0,such aφis called singular,N≥1,n≥3 are integers,tk are the grid points,uk:=u(tk),k=1,2,...,n,∇is the backward difference operator defined by∆uk=uk-uk-1,△is the forward difference operator defined by△uk=uk+1-uk,fk(2≤k≤n-1)are continuous functions.We prove the existence of solutions to this problem by employing the sign condition,the continuation lemma and the upper and lower solutions,respectively.On this basis,we also establish the Ambrosetti-Prodi type results for it.展开更多
This paper presents new existence results for singular discrete boundary value problems. In particular our nonlinearity may be singular in its dependent variable and is allowed to change sign.
By means of the critical point theory,we prove the existence of positive solutions for a higher dimensional discrete boundary value problem.Our results generalize one of Agarwal in[2].
In this paper,using the critical point theory,we obtain a new result on the existence of the solutions to a class of n-dimensional discrete boundary value problems.Results obtained extend or improve the existing ones.
In this paper, we consider the existence of three nontrivial solutions for a discrete non-linear multiparameter periodic problem involving the p-Laplacian. By using the similar method for the Dirichlet boundary value ...In this paper, we consider the existence of three nontrivial solutions for a discrete non-linear multiparameter periodic problem involving the p-Laplacian. By using the similar method for the Dirichlet boundary value problems in [C. Bonanno and P. Candito, Appl. Anal., 88（4） （2009）, pp. 605-616], we construct two new strong maximum principles and obtain that the boundary value problem has three positive solutions for λ and μ in some suitable intervals. The approaches we use are the critical point theory.展开更多
In his series of three papers we study singularly perturbed (SP) boundary valueproblems for equations of elliptic and parabolic type. For small values of the pertur-bation parameter parabolic boundary and interior lay...In his series of three papers we study singularly perturbed (SP) boundary valueproblems for equations of elliptic and parabolic type. For small values of the pertur-bation parameter parabolic boundary and interior layers appear in these problems.If classical discretisation methods are used, the solution of the finite differencescheme and the approximation of the diffusive flux do not converge uniformly withrespect to this parameter. Using the method of special, adapted grids, we canconstruct difference schemes that allow approximation of the solution and the nor-malised diffusive flux uniformly with respect to the small parameter.We also consider sillgularly perturbed boundary value problems for convection-diffusion equations. Also for these problems we construct special finite differenceschemes, the solution of which converges ε-uniformly We study what problems ap-pear, when classical schemes are used for the approximation of the spatial deriva-tives. We compare the results with those obtained by the adapted approach. Re-sults of numerical experiments are discussed.In the three papers we first give an introduction on the general problem, andthen we consider respectively (i) Problems for SP parabolic equations, for whichthe solution and the normalised diffusive fluxes are required; (ii) Problems for SPelliptic equations with boundary conditions of Diriclilet, Neumann and RDbin type;(iii) Problems for SP parabolic equation with discontinuous boundary conditions-展开更多
In this series of three papers we study singularly perturbed (SP) boundary vaue problems for equations of elliptic and parabolic troe. For small values of the perturbation parameter parabolic boundary and interior lay...In this series of three papers we study singularly perturbed (SP) boundary vaue problems for equations of elliptic and parabolic troe. For small values of the perturbation parameter parabolic boundary and interior layers appear in these problems. If classical discretisation methods are used, the solution of the finite difference scheme and the approximation of the diffusive flux do not converge uniformly with respect to this parameter. Using the method of special, adapted grids,we can construct difference schemes that allow approkimation of the solution and the normalised diffusive flux uniformly with respect to the small parameter.We also consider singularly perturbed boundary value problems for convection diffusion equations. Also for these problems we construct special finite difference schemes, the solution of which converges E-uniformly We study what problems appear, when classical schemes are used for the approximation of the spatial deriva tives. We compare the results with those obtained by the adapted approach. Results of numerical experiments are discussed.In the three papers we first give an introduction on the general problem, and then we consider respectively (i) Problems for SP parabolic equations, for which the solution and the normalised diffusive fluxes are required; (ii) Problems for SP elliptic equations with boundary conditions of Dirichlet, Neumann and Robin type;(iii) Problems for SP parabolic eqllation with discontinuous boundaxy conditions展开更多
In this paper we establish the existence of single and multiple positive solutions to the following singular discrete boundary value problem {△[Ф△x（i-1））]＋q1（i）f1（i,x（i）,y（i））=0,i∈{1,2,…,T}△[Ф△x（...In this paper we establish the existence of single and multiple positive solutions to the following singular discrete boundary value problem {△[Ф△x（i-1））]＋q1（i）f1（i,x（i）,y（i））=0,i∈{1,2,…,T}△[Ф△x（i-1））]＋q1（i）f2（i,x（i）,y（i））=0,x（0）=x（T＋1）=y（T＋1）=y（0）=y（T＋1）=0,whereФ（s）=｜s｜^p-2s,p〉1 and the nonlinear terms fk（i,x,y）（k=1,2）may be singular at （x,y）=（0,0）.展开更多
基金Supported by the National Natural Science Foundation of China(Grant Nos.1236104012461035)+1 种基金the Outstanding Youth Fund of Gansu Province(Grant No.24JRRA121)the Scientific Research Ability Improvement Program for Young Teachers of Northwest Normal University(Grant No.NWNU-LKQN2021-17)。
文摘In this paper,we consider the discrete boundary value problem of the type{∆u1=0=∆un-1,∇(t_(k)^(N-1))φ(∆uk))+t_(k)^(N-1)fk(t_(k),u_(k),∆_(uk))=0,2≤k≤n-1,whereφ:(-a,a)→R,0<a<∞,is an increasing homeomorphism withφ(0)=0,such aφis called singular,N≥1,n≥3 are integers,tk are the grid points,uk:=u(tk),k=1,2,...,n,∇is the backward difference operator defined by∆uk=uk-uk-1,△is the forward difference operator defined by△uk=uk+1-uk,fk(2≤k≤n-1)are continuous functions.We prove the existence of solutions to this problem by employing the sign condition,the continuation lemma and the upper and lower solutions,respectively.On this basis,we also establish the Ambrosetti-Prodi type results for it.
文摘This paper presents new existence results for singular discrete boundary value problems. In particular our nonlinearity may be singular in its dependent variable and is allowed to change sign.
基金This work is supported by Scientific Research Plan Item of Hunan Provincial Department of Education(No.04C491).
文摘By means of the critical point theory,we prove the existence of positive solutions for a higher dimensional discrete boundary value problem.Our results generalize one of Agarwal in[2].
基金supported by the Foundation of Education Department of Guangxi Province(No.200911LX427)
文摘In this paper,using the critical point theory,we obtain a new result on the existence of the solutions to a class of n-dimensional discrete boundary value problems.Results obtained extend or improve the existing ones.
基金Supported by NSFC(11326127,11101335)NWNULKQN-11-23the Fundamental Research Funds for the Gansu Universities
文摘In this paper, we consider the existence of three nontrivial solutions for a discrete non-linear multiparameter periodic problem involving the p-Laplacian. By using the similar method for the Dirichlet boundary value problems in [C. Bonanno and P. Candito, Appl. Anal., 88（4） （2009）, pp. 605-616], we construct two new strong maximum principles and obtain that the boundary value problem has three positive solutions for λ and μ in some suitable intervals. The approaches we use are the critical point theory.
文摘In his series of three papers we study singularly perturbed (SP) boundary valueproblems for equations of elliptic and parabolic type. For small values of the pertur-bation parameter parabolic boundary and interior layers appear in these problems.If classical discretisation methods are used, the solution of the finite differencescheme and the approximation of the diffusive flux do not converge uniformly withrespect to this parameter. Using the method of special, adapted grids, we canconstruct difference schemes that allow approximation of the solution and the nor-malised diffusive flux uniformly with respect to the small parameter.We also consider sillgularly perturbed boundary value problems for convection-diffusion equations. Also for these problems we construct special finite differenceschemes, the solution of which converges ε-uniformly We study what problems ap-pear, when classical schemes are used for the approximation of the spatial deriva-tives. We compare the results with those obtained by the adapted approach. Re-sults of numerical experiments are discussed.In the three papers we first give an introduction on the general problem, andthen we consider respectively (i) Problems for SP parabolic equations, for whichthe solution and the normalised diffusive fluxes are required; (ii) Problems for SPelliptic equations with boundary conditions of Diriclilet, Neumann and RDbin type;(iii) Problems for SP parabolic equation with discontinuous boundary conditions-
文摘In this series of three papers we study singularly perturbed (SP) boundary vaue problems for equations of elliptic and parabolic troe. For small values of the perturbation parameter parabolic boundary and interior layers appear in these problems. If classical discretisation methods are used, the solution of the finite difference scheme and the approximation of the diffusive flux do not converge uniformly with respect to this parameter. Using the method of special, adapted grids,we can construct difference schemes that allow approkimation of the solution and the normalised diffusive flux uniformly with respect to the small parameter.We also consider singularly perturbed boundary value problems for convection diffusion equations. Also for these problems we construct special finite difference schemes, the solution of which converges E-uniformly We study what problems appear, when classical schemes are used for the approximation of the spatial deriva tives. We compare the results with those obtained by the adapted approach. Results of numerical experiments are discussed.In the three papers we first give an introduction on the general problem, and then we consider respectively (i) Problems for SP parabolic equations, for which the solution and the normalised diffusive fluxes are required; (ii) Problems for SP elliptic equations with boundary conditions of Dirichlet, Neumann and Robin type;(iii) Problems for SP parabolic eqllation with discontinuous boundaxy conditions
基金Supported by the XJZDXK of China(Grant No.XJZDXK2011004)the National Natural Science Foundation of China(Grant No.10971021)
文摘In this paper we establish the existence of single and multiple positive solutions to the following singular discrete boundary value problem {△[Ф△x（i-1））]＋q1（i）f1（i,x（i）,y（i））=0,i∈{1,2,…,T}△[Ф△x（i-1））]＋q1（i）f2（i,x（i）,y（i））=0,x（0）=x（T＋1）=y（T＋1）=y（0）=y（T＋1）=0,whereФ（s）=｜s｜^p-2s,p〉1 and the nonlinear terms fk（i,x,y）（k=1,2）may be singular at （x,y）=（0,0）.
